Overview

Permutations and combinations are the basic tools of counting.

Many contest problems ask us to count the number of ways to choose, arrange, or assign objects. The main difficulty is usually deciding whether order matters.

If order matters, we usually use permutations. If order does not matter, we usually use combinations.

This module introduces the two most important counting expressions:

and

These are often written as $nPr$ and $nCr$.

Motivation

Suppose there are $5$ students:

If we want to choose $2$ students for a committee, then the order does not matter. Choosing $A$ and $B$ is the same as choosing $B$ and $A$.

So this is a combination problem.

However, if we want to choose a president and vice-president, then order does matter. Choosing $A$ as president and $B$ as vice-president is different from choosing $B$ as president and $A$ as vice-president.

So this is a permutation problem.

The first question in almost every counting problem should be:

Does order matter?

Factorials

For a positive integer $n$, define

For example,

We also define

This may seem strange at first, but it makes many counting formulas work cleanly.

Factorials appear whenever we arrange distinct objects.

For example, the number of ways to arrange $5$ distinct books in a row is

Permutations

A permutation is an ordered selection.

The number of ways to choose and arrange $r$ objects from $n$ distinct objects is

This formula appears because we have:

• $n$ choices for the first position
• $n-1$ choices for the second position
• $n-2$ choices for the third position and so on

As such

Example 1

How many ways are there to choose a president, vice-president, and secretary from $8$ students?

The roles are distinct, so order matters.

We need to arrange $3$ students chosen from $8$:

So there are $336$ possible choices.

Combinations

A combination is an unordered selection.

The number of ways to choose $r$ objects from $n$ distinct objects is

This counts selections where order does not matter.

Example 2

How many ways are there to choose $3$ students from $8$ students for a committee?

The committee has no roles, so order does not matter.

Therefore the answer is

So there are $56$ possible committees.

Relationship Between Permutations and Combinations

Permutations and combinations are closely related.

First choose $r$ objects from $n$ objects:

Then arrange those $r$ chosen objects:

Therefore,

This identity is useful because many problems can be solved by thinking:

First choose the objects, then arrange them.

Choosing Then Arranging

A very common counting strategy is:

1. Choose the objects involved
2. Arrange the chosen objects

Example 3

How many $4$-letter arrangements can be made from the letters $A,B,C,D,E,F$ if no letter is repeated?

First choose the $4$ letters:

Then arrange them:

So the number of arrangements is

This is the same as

When Repetition Is Allowed

If repetition is allowed, the formulas change.

For example, the number of $4$-digit strings using digits $0$ through $9$, where repetition is allowed, is

This is because each of the $4$ positions has $10$ choices.

If the first digit cannot be $0$, then there are:

• $9$ choices for the first digit
• $10$ choices for each of the remaining three digits

So the number of $4$-digit positive integers allowing repeated digits is

Worked Examples

Example 4

How many ways are there to arrange $5$ people in a row?

All $5$ people are distinct, and order matters.

The answer is

Example 5

How many ways are there to choose $4$ cards from a standard $52$ card deck?

The order in which the cards are chosen does not matter.

So the answer is

Example 6

How many ways are there to choose and arrange $4$ runners from $10$ runners into first, second, third, and fourth place?

The positions are distinct, so order matters.

The answer is

Example 7

A club has $12$ members. How many ways are there to choose $3$ members for a committee and then choose one of those committee members as chair?

First choose the committee:

Then choose the chair from the $3$ committee members:

So the total number of ways is

Since

the answer is

Key Contest Insight

The most important decision is whether order matters.

If order matters, use permutations or arrange objects directly.

If order does not matter, use combinations.

If a problem involves both choosing and assigning roles, separate the process into steps:

• Choose the objects
• Then arrange or assign them

This helps prevent overcounting.

Strategy Tips

• Ask whether order matters
• Use factorials for arranging distinct objects
• Use $nPr$ for ordered selections without repetition
• Use $nCr$ for unordered selections
• If repetition is allowed, count choices position by position
• Break complicated problems into smaller choices
• Be careful not to count the same outcome multiple times

Common Pitfalls

• Using $nPr$ when order does not matter
• Using $nCr$ when order does matter
• Forgetting that repeated objects require different handling
• Treating a committee as ordered when it is unordered
• Forgetting restrictions such as no leading zero
• Multiplying choices without checking whether earlier choices affect later choices

Practice Problems